Skip to main content
SpringerLink
Log in

On the Computation of Recurrence Coefficients for Univariate Orthogonal Polynomials

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Associated to a finite measure on the real line with finite moments are recurrence coefficients in a three-term formula for orthogonal polynomials with respect to this measure. These recurrence coefficients are frequently inputs to modern computational tools that facilitate evaluation and manipulation of polynomials with respect to the measure, and such tasks are foundational in numerical approximation and quadrature. Although the recurrence coefficients for classical measures are known explicitly, those for nonclassical measures must typically be numerically computed. We survey and review existing approaches for computing these recurrence coefficients for univariate orthogonal polynomial families and propose a novel “predictor–corrector” algorithm for a general class of continuous measures. We combine the predictor–corrector scheme with a stabilized Lanczos procedure for a new hybrid algorithm that computes recurrence coefficients for a fairly wide class of measures that can have both continuous and discrete parts. We evaluate the new algorithms against existing methods in terms of accuracy and efficiency.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Barkov, G.I.: Some systems of polynomials orthogonal in two symmetric intervals. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika 4, 3–16 (1960)

    MathSciNet  MATH  Google Scholar 

  2. Chebyshev, P..L.: Sur l’interpolation par la méthode des moindres carrés. Mémoires de lÁcadémie Impériale des sciences de St.-Pétersbourg 1(15), 1–24 (1859)

  3. Chihara, T.S.: An introduction to orthogonal polynomials. Courier Corporation (2011)

  4. Freud, G.: Orthogonal Polynomials. Pergamon Press (1971)

  5. Freud, G.: On the coefficients in the recursion formulae of orthogonal polynomials. In: Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, pp. 1–6. JSTOR (1976)

  6. Gautschi, W.: https://www.cs.purdue.edu/archives/2002/wxg/codes/sr_freud.m

  7. Gautschi, W.: A survey of gauss-christoffel quadrature formulae, em “eb christoffel-the influence of his work in mathematics and physical sciences"(pl butzer e f. fehér, eds.) pp. 72-147 (1981). https://doi.org/10.1007/978-3-0348-5452-8_6

  8. Gautschi, W.: On generating orthogonal polynomials. SIAM J. Sci. Stat. Comput. 3(3), 289–317 (1982). https://doi.org/10.1137/0903018

    Article  MathSciNet  MATH  Google Scholar 

  9. Gautschi, W.: On some orthogonal polynomials of interest in theoretical chemistry. BIT Numer. Math. 24(4), 473–483 (1984)

    Article  MathSciNet  Google Scholar 

  10. Gautschi, W.: Algorithm 726: ORTHPOL—a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Softw. 20(1), 21–62 (1994). https://doi.org/10.1145/174603.174605

    Article  MATH  Google Scholar 

  11. Gautschi, W.: Algorithm 726: Orthpol-a package of routines for generating orthogonal polynomials and gauss-type quadrature rules. ACM Trans. Math. Softw. (TOMS) 20(1), 21–62 (1994)

    Article  Google Scholar 

  12. Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Oxford University Press, USA (2004)

    Book  Google Scholar 

  13. Gautschi, W.: Orthogonal polynomials, quadrature, and approximation: computational methods and software (in Matlab). In: Marcellán, F., Assche, W. V. (eds.) Orthogonal Polynomials and Special Functions, no. 1883 in Lecture Notes in Mathematics, pp. 1–77. Springer, Heidelberg (2006). https://doi.org/10.1007/978-3-540-36716-1_1

  14. Gautschi, W.: Variable-precision recurrence coefficients for nonstandard orthogonal polynomials. Numer. Algorithms 52(3), 409–418 (2009)

    Article  MathSciNet  Google Scholar 

  15. Glaws, A., Constantine, P.G.: Gaussian quadrature and polynomial approximation for one-dimensional ridge functions. SIAM J. Sci. Comput. 41(5), S106–S128 (2019)

    Article  MathSciNet  Google Scholar 

  16. Gragg, W.B., Harrod, W.J.: The numerically stable reconstruction of jacobi matrices from spectral data. Numerische Mathematik 44(3), 317–335 (1984)

    Article  MathSciNet  Google Scholar 

  17. https://www.mathworks.com/help/symbolic/vpa.html

  18. https://github.com/ZEXINLIU/Univariate_ttr_examples

  19. Lew, J.S., Quarles, D.A., Jr.: Nonnegative solutions of a nonlinear recurrence. J. Approx. Theory 38(4), 357–379 (1983)

    Article  MathSciNet  Google Scholar 

  20. Lubinsky, D.S., Mhaskar, H.N., Saff, E.B.: A proof of Freud’s conjecture for exponential weights. Constr. Approx. 4(1), 65–83 (1988). https://doi.org/10.1007/BF02075448

  21. Magnus, A.P.: Freud’s equations for orthogonal polynomials as discrete painlevé equations. arXiv:math/9611218 pp. 7–8 (1996)

  22. Nevai, P.G.: Orthogonal Polynomials. American Mathematical Society (1980)

  23. Oladyshkin, S., Nowak, W.: Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion. Reliab. Eng. Syst. Saf. 106, 179–190 (2012). https://doi.org/10.1016/j.ress.2012.05.002

    Article  Google Scholar 

  24. Pinkus, A.: Ridge Functions, vol. 205. Cambridge University Press (2015)

  25. Rutishauser, H.: On Jacobi rotation patterns. Proc. Symp. Appl. Math. 15, 219–239 (1963)

    Article  MathSciNet  Google Scholar 

  26. Sack, R.A., Donovan, A.F.: An algorithm for Gaussian quadrature given modified moments. Numerische Mathematik 18(5), 465–478 (1971). https://doi.org/10.1007/BF01406683

    Article  MathSciNet  MATH  Google Scholar 

  27. Smith, R..C.: Uncertainty Quantification: Theory, Implementation, and Applications, vol. 12. Siam (2013)

  28. Stieltjes, T.J.: Quelques recherches sur la théorie des quadratures dites mécaniques. Annales scientifiques de l’École Normale Supérieure 1, 409–426 (1884)

  29. Stieltjes, T.J.: Some research on the theory of so-called mechanical quadratures. Scientific annals of the ’Ecole Normale Sup é rieure 1, 409–426 (1884)

  30. Sullivan, T.J.: Introduction to Uncertainty Quantification, vol. 63. Springer (2015)

  31. Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Soc (1975)

  32. Van Assche, W.: Discrete Painlevé equations for recurrence coefficients of orthogonal polynomials. In: Difference Equations, Special Functions and Orthogonal Polynomials, pp. 687–725. World Scientific (2007)

  33. Wheeler, J.C.: Modified moments and Gaussian quadratures. Rocky Mountain J. Math. 4(2), 287–296 (1974). https://doi.org/10.1216/RMJ-1974-4-2-287

    Article  MathSciNet  MATH  Google Scholar 

  34. Wheeler, J.C.: Modified moments and continued fraction coefficients for the diatomic linear chain. J. Chem. Phys. 80(1), 472–476 (1984)

    Article  Google Scholar 

  35. Wiener, N.: The homogeneous chaos. Am. J. Math. 60(4), 897–936 (1938)

    Article  MathSciNet  Google Scholar 

  36. Witteveen, J.A., Bijl, H.: Modeling arbitrary uncertainties using Gram–Schmidt polynomial chaos. In: 44th AIAA Aerospace Sciences Meeting and Exhibit, p. 896 (2006)

  37. Witteveen, J.A., Sarkar, S., Bijl, H.: Modeling physical uncertainties in dynamic stall induced fluid-structure interaction of turbine blades using arbitrary polynomial chaos. Comput. Struct. 85(11–14), 866–878 (2007)

    Article  Google Scholar 

  38. Xiu, D., Karniadakis, G.E.: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002). https://doi.org/10.1137/S1064827501387826

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Institute of Biomedical Imaging and Bioengineering of the National Institutes of Health under grant number U24EB029012, and under National Science Foundation awards DMS-1720416 and DMS-1848508. This material is based upon work supported by both the National Science Foundation under Grant No. DMS-1439786 and the Simons Foundation Institute Grant Award ID 507536 while A. Narayan was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Spring 2020 semester

Author information

Authors and Affiliations

  1. Department of Mathematics, and Scientific Computing and Imaging (SCI) Institute, The University of Utah, Salt Lake City, USA

    Zexin Liu & Akil Narayan

Authors
  1. Zexin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Akil Narayan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zexin Liu.

Ethics declarations

Conflicts of interest

The authors have no conflicts of interest to declare that are relevant to the content of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, Z., Narayan, A. On the Computation of Recurrence Coefficients for Univariate Orthogonal Polynomials. J Sci Comput 88, 53 (2021). https://doi.org/10.1007/s10915-021-01586-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01586-w

Keywords

Mathematics Subject Classification

Search

Navigation